Georgian Mathematical Journal 1(1994), No. 2, 151-172

ON COHOMOTOPY-TYPE FUNCTORS

S. KHAZHOMIA

Abstract. This article deals with Chogoshvili cohomotopy functors which are defined by extending a cohomology functor given on some special auxiliary subcategories of the category of topological spaces. The question of choosing these subcategories is discussed. In particular, it is shown that in the singular case to define absolute groups it is sufficient that auxiliary subcategories should have as objects only spheres S^{n}, Moore spaces P^{n}(t) = S^{n−1} ∪_{t} e^{n}, and one-point unions of these spaces.

In [2, 3] for any cohomology theory H = {H^{n}} given on some category K of pairs of topological spaces the sequence Π = {Π^{n}}, n = 0, 1, 2, …, of contravariant functors Π^{n} is constructed from K into the category of abelian groups with the coboundary operator δ^{#} which commutes with the induced homomorphisms φ^{#}, φ ∈ K. Functors Π^{n} possess the properties of semi-exactness and homotopy and are connected with H by the natural transformations d : H^{n} → Π^{n} which are the natural equivalences on a certain subcategory K_{n} of K. Constructing such functors is reduced to the problem of extending the functor given on an auxiliary subcategory K_{n} to the whole category K. The problem is solved by means of the theory of inverse systems of groups with sets of homomorphisms of Hurewiz, Dugundji, and Dowker [4].

Functors Π^{n} are dual to the homotopy functors associated in the sense of Bauer [1] with a given homological structure. It should be noted that functors Π^{n} have some of the basic properties of the Borsuk cohomotopy, but they differ from the latter.

In [5-7] functors Π^{n} were investigated under the assumption that K is the category of pairs of topological spaces with a base point and base point preserving maps, and H is the singular integral theory of cohomology. We will adhere to the same assumption throughout this paper (base points are not indicated here). To define functors Π^{n} we need auxiliary subcategories K_{n}. We consider the problem of choosing these subcategories.

1991 Mathematics Subject Classification. 55Q55.

For convenience we recall the definition of a limit of the inverse system of groups with sets of homomorphisms (see [4]). Let ω be a partially ordered set, and {G_{σ}} a system of abelian groups indexed by the elements of ω. Furthermore, let, for each pair ρ < σ, sets H_{σρ} ⊂ Hom (G_{σ}, G_{ρ}) be given such that if ρ < σ < τ and φ_{1} ∈ H_{σρ}, φ_{2} ∈ H_{τσ}, then the composition φ_{1}φ_{2} ∈ H_{τρ}. Then, by definition, lim_{←} G_{σ} is a subgroup of the group ΠG_{σ} and its elements are elements g = {g_{σ}} ∈ ΠG_{σ} such that for each pair ρ < σ and φ ∈ H_{σρ} we have φ(g_{σ}) = g_{ρ}.

It should be noted that this theory of [4] is essentialy Kan’s extension theory in its early stage, but quite sufficient for our purpose.

The results of this paper were announced earlier in [6,7].

1. Preliminaries

In this section we will give the definitions of subcategories K_{n} and functors Π^{n} and discuss some of their properties.

Let e^{m} be the unit m-cell of the m-dimensional euclidean space ℝ^{m}. By e^{0} we denote some fixed point (base point). Let K̃ be the small full subcategory of K whose objects are all finite CW -complexes X for which X^{0} = e^{0} and X^{k} is the adjunction space obtained by adjoining a finite

number of e^{k} to X^{k−1}, k > 0. We denote by ˜ K̃ the small full subcategory of K whose objects are all CW -pairs (X, X′) for which X and X′ are the objects of K̃.

Now we shall define auxiliary subcategories K_{n}, n > 3, by the following two conditions (cf. [2, 5]):

1) K_{n} is an arbitrary small full subcategory of K; each object of K_{n} is a pair (X, X′) of linearly and simply connected spaces satisfing the conditions that π_{2}(X, X′) = 1, the homology modules H_{∗}(X) and H_{∗}(X′) are of finite type, H^{i}(X, X′) = 0, i < n, and H^{i}(X)= H^{i}(X′) = 0, 0 < i < n − 1;

2) K_{n} contains all possible objects of ˜ K̃.

We denote by F_{n}^{r} an auxiliary subcategory of objects only of ˜ K̃.

If n = 3, we assume that K_{3} is an arbitrary (containing all possible objects of K̃) small full subcategory of K whose all objects are linearly, and simply connected spaces X for which H_{∗}(X) is a module of finite type and H^{2}(X) = 0.

Let (R, R′) be an object of K. Consider a set of indices ω(R, R′; n) of all pairs α = (X, X′; f), where (X, X′) is an object of K_{n} and f : (X, X′) → (R, R′) is a continuous map of K. Let ω(R, R′; n) be ordered as follows: α < β, where β = (Y, Y′; g) if there is a map φ : (X, X′) → (Y, Y′) of K_{n} such that

gφ = f.  (1)

Assume that to every α ∈ ω(R, R′; n) there corresponds the n-dimensio-nal cohomology group H_{α} = H^{n}(X, X′) and to every ordered pair α < β there corresponds the set of homomorphisms {φ^{∗}}, where φ^{∗} : H^{n}(Y, Y′) → H^{n}(X, X′) are the induced homomorphisms in the H theory.

We have obtained the inverse system of the group H_{α} with sets of homomorphisms. Cohomotopic groups of Chogoshvili are determined by the formula Π^{n}(R, R′; K_{n}) = lim_{←} H_{α}.

We denote by Π^{n}(R; K_{n}) the absolute group Π^{n}(R, ∗; K_{n}), where ∗ is a base point, and by p_{α} the α-coordinate of an element p ∈ Π^{n}(R, R′; K_{n}). Note that for n = 3 we have determined the absolute groups only.

Let

α = (X, X′; f), β = (X, X′; g),
α, β ∈ ω(R, R′; n), p ∈ Π^{n}(R, R′; K_{n})

and let the maps f and g be homotopic, i.e., f ∼ g. Let I be the unit segment.

Lemma 1.1. If the subcategory K_{n} contains, alongside with (X, X′), the pair (X × I, X′ × I), then p_{α} = p_{β}.

Proof. See [5], p. 83.

Let

α = (X, X′; f), β = (Y, Y′; g),
α, β ∈ ω(R, R′; n), p ∈ Π^{n}(R, R′; K_{n}).

Moreover, let us have a map φ : (X, X′) → (Y, Y′) from K_{n} such that the maps gφ and f are homotopic, i.e.,

gφ ∼ f.  (2)

Lemma 1.2. If the subcategory K_{n} contains, alongside with (X, X′), the pair (X × I, X′ × I), then φ^{∗}(p_{β}) = p_{α}.

Proof. Consider the index (X, X′; gφ) = α_{1} < β and apply Lemma 1.1 to α_{1} and α.

Let α=(X, X′; f), where f is null-homotopic, and p∈Π^{n}(R, R′; K_{n}).

Corollary 1.3. If the subcategory K_{n} contains, alongside with (X, X′), the pair (X × I, X′ × I), then p_{α} = 0.

Proof. Apply Lemma 1.2 to the homotopy commutative diagram

[figure: (X, X′); (X, X′) ,; (R, R′); f; o; o]

where o denotes the constant map.

2. Main Results

Let K′_{n} be a subcategory of K′′_{n}, where K′_{n} and K′′_{n} are two auxiliary subcategories, and let (R, R′) ∈ K, p ∈ Π^{n}(R, R′; K′′_{n}), α ∈ ′ω(R, R′; n) ⊂ ′′ω(R, R′; n). Then, as one can easily verify, the formula [λ(p)]_{α} = p_{α} defines the restriction homomorphism λ : Π^{n}(R, R′; K′′_{n}) → Π^{n}(R, R′; K′_{n}).

In Section 3 we will prove

Theorem 2.1. The homomorphism λ defines the natural equivalence of the functors Π^{n}(−, −; K′′_{n}) and Π^{n}(−, −; K′_{n}), n > 3. In particular, all functors Π^{n}(−, −; K_{n}) are naturally equivalent to the functor Π^{n}(−, −; F_{n}^{r}).

Remark 2.2. Theorem 2.1 shows that in choosing a subcategory K_{n} we can restrict ourselves only to the finite CW -pairs. On the other hand, from Theorem 2.1 it follows that for the convenience of construction and proof we can regard an arbitrary admissible pair as an object of K_{n}.

Remark 2.3. One can easily show that Theorem 2.1 holds for the absolute groups when n > 2. Moreover, in defining the absolute groups, to choose the subcategory K_{n} we can restrict ourselves to the absolute pairs (X, ∗), i.e., X′ = ∗ (see the definition of K_{n} and [5]).

In the remainder of this section we will consider the absolute groups only. Therefore to define the functors Π^{n}, n > 2, we can use auxiliary subcategories F_{n}^{a} consisting of finite CW -complexes. More exactly, F_{n}^{a} is a full subcategory K̃ whose objects are all spaces X for which π_{1}(X) = 1 and H^{2}(X) = ⋯ = H^{n−1}(X) = 0.

We intend here to study the problem dealing with the possibility of further reducing subcategories K_{n} provided that groups Π^{n}(R; K_{n}) and the results from [5-7] remain unchanged. To this effect, relying on Lemma 1.2, in the definition of Π^{n}(R; K_{n}) we replace condition (1) by condition (2) (see Section 1). We will stick to this definition in the sequel.

Let S^{n} denote the n-dimensional unit sphere of the euclidean space ℝ and e^{n} the unit disk. Denote by P^{n}(t), t > 1, n > 2, the Moore space S^{n−1} ∪_{t} e^{n}. Also assume that P^{n}(1) = S^{n}.

Consider now the full subcategory F_{n}^{b} of F_{n}^{a} whose objects are all finite one-point unions of spaces P^{n}(t), t ≥ 1. The subcategory F_{n}^{b} will be regarded as an auxiliary subcategory.

The following theorem will be proved in Section 4.

Theorem 2.4. The restriction homomorphism defines the natural equivalence of the functors Π^{n}(−; F_{n}^{a}) and Π^{n}(−; F_{n}^{b}), n > 2.

We introduce the following notations:

1) P_{j}^{n}(t) = P^{n}(t), where j is a positive integer, n > 2, t ≥ 1;

2) X_{k}^{n} = ∨_{t=1}^{k}( ∨_{j=1}^{k} P_{j}^{n}(t));

3) Q^{n} = lim_{→} X_{k} (by inclusion maps X_{k}^{n} → X_{k+1}^{n}).

Let Q_{n} be the full subcategory of K consisting of one object Q^{n}, n > 2. The subcategory will also be regarded as an auxiliary subcategory. Note that the module H_{∗}(Q^{n}) is not obviously of the finite type. Therefore none of the above-defined auxiliary subcategories contains Q^{n}.

The following theorem will be proved in Section 5.

Theorem 2.5. The functors Π^{n}(−; F_{n}^{b}) and Π^{n}(−; Q_{n}) are naturally equivalent, n > 2.

3. Proof of Theorem 2.1

Let us prove that λ is natural. Assume f : (S, S′) → (R, R′) to be an arbitrary map of K. Consider the diagram

[figure: (M, M′); (N, N′); (R, R′); φ; f; fφ^{−1}]

Let

α = (X, X′; g) ∈ ′ω(S, S′, n) ⊂ ′′ω(S, S′; n),
β = f(α) = (X, X′; fg) ∈ ′ω(R, R′; n) ⊂ ′′ω(R, R′; n)

and let p ∈ Π^{n}(R, R′; K′′_{n}). We have

[λ(f^{#}(p))]_{α} = [f^{#}(p)]_{α} = p_{β},
[f^{#}(λ(p))]_{α} = [λ(p)]_{β} = p_{β},

which proves that λ is natural.

Let (X, X′) be an arbitrary object of some auxiliary subcategory K_{n}. Using the standard technique of the homotopy theory, we can construct a CW -pair (A, B) from the subcategory K_{n} and a map φ : (A, B) → (X, X′) such that homomorphisms φ^{∗} induced by φ in the H theory will be isomorphisms up to any pregiven dimension. Let now p ∈ Π^{n}(R, R′; K_{n}) and α = (X, X′; f) ∈ ω(R, R′; n). Consider the index β = (A, B; fφ) ∈ ω(R, R′; n). Then β < α and therefore p_{β} = φ^{∗}(p_{α}). Hence p_{α} = φ^{∗−1}(p_{β}). From the above reasoning and the definition of auxiliary subcategories it now follows that if p ∈ Π^{n}(R, R′; K′′_{n}) and λ(p) = 0, then p = 0. Thus λ is a monomorphism.

Let L_{n}, n > 3, be a full subcategory of the category K whose objects are all pairs (X, X′) for which X and X′ are linearly and simply connected spaces, π_{2}(X, X′) = 1, the homology modules H_{∗}(X) and H_{∗}(X′) are of the finite type, H^{i}(X, X′) = 0, i < n, and H^{i}(X) = H^{i}(X′) = 0, 0 < i < n − 1.

Consider some full subcategories of L_{n}:

1) L_{n}^{(1)} are CW -pairs;

2) L_{n}^{(2)} are CW -pairs with a finite number of cells in all demensions;

3) L_{n}^{(3)} are finite CW -pairs;

4) L_{n}^{(4)} = ˜ K̃ ∩ Ł_{n}^{(3)} = F_{n}^{r}.

Wewillgraduallyextendthethreaddefiningelementp∈Π^{n}(R, R′; F_{n}^{r}) from the category L_{n}^{(4)} onto L_{n}^{(3)}, then onto L_{n}^{(2)}, L_{n}^{(1)} and, finally, onto L_{n}. Such an extension already implies that λ is epimorphic.

Let (M, M′) be an arbitrary object of L_{n}^{(3)}; also let f : (M, M′) → (R, R′) be an arbitrary map. Consider the diagram

[figure: Π^{n}(S, S′; K′′_{n}); Π^{n}(S, S′; K′_{n}); Π^{n}(R, R′; K′_{n}); Π^{n}(R, R′; K′′_{n}); λ; λ; f^{#}; f^{#}]

where (N, N′) ∈ L_{n}^{(4)} and φ is a homeomorphism. Let α = (M, M′; f) and β = (N, N′; fφ^{−1}). It is assumed that p_{α} = φ^{∗}(p_{β}). We will show that p_{α} does not depend on the choice of the homeomorphism φ. Consider the diagram

[figure: (N, N′); (N̄, N̄′); (M, M′); (R, R′); g; φ_{1}; φ_{2}; fφ_{1}^{−1}; fφ_{2}^{−1}; f]

where φ_{1} and φ_{2} are two different homeomorphisms and g = φ_{2}φ_{1}^{−1}. The indices β_{1} and β_{2} will be defined similarly to β. We have (fφ_{2}^{−1})g = fφ_{2}^{−1}φ_{2}φ_{1}^{−1} = fφ_{1}^{−1}. Therefore β_{1} < β_{2}. Then φ_{2}^{∗}(p_{β_{2}}) = (gφ_{1})^{∗}(p_{β_{2}}) = φ_{1}^{∗}(g^{∗}(p_{β_{2}})) = φ_{1}^{∗}(p_{β_{1}}).

Consider the commutative diagram

[figure: (N, N′); (N_{0}, N′_{0}); (M, M′); (M_{0}, M′_{0}); (R, R′); g_{0}; g; φ; φ_{0}; f; f_{0}]

where φ and φ_{0} are homeomorphisms, g_{0} = φ_{0}gφ^{−1}.

Let

α = (M, M′; f), α_{0} = (M_{0}, M′_{0}; f_{0}),
β = (N, N′; fφ^{−1}), β_{0} = (N_{0}, N′_{0}; f_{0}φ_{0}^{−1}).

We have

(f_{0}φ_{0}^{−1})g_{0} = f_{0}φ_{0}^{−1}φ_{0}gφ^{−1} = f_{0}gφ^{−1} = fφ^{−1}.

Therefore β < β_{0}. Then

g^{∗}(p_{α_{0}}) = g^{∗}(φ_{0}^{∗}(p_{β_{0}})) = (φ_{0}g)^{∗}(p_{β_{0}}) = (g_{0}φ)^{∗}(p_{β_{0}}) =
= φ^{∗}(g_{0}^{∗}(p_{β_{0}})) = φ^{∗}(p_{β}) = p_{α}.

We have thus extended the thread of the element p onto the category L_{n}^{(3)}.

Let, now, (M, M′) ∈ L_{n}^{(2)} and f : (M, M′) → (R, R′) be an arbitrary map. By i : X_{k} → X we denote here the standard embedding, where X_{k} is the k-skeleton of the CW -complex X. Let N_{1} > N > n + 1 be arbitrary integers.

Consider the commutative diagram

[figure: (M_{N}, M′_{N}); (M_{N_{1}}, M′_{N_{1}}); (M, M′); (R, R′); i; i_{1}; f_{N_{1}}; f_{N}; f]

where f_{N_{1}} = fi_{1} and f_{N} = f_{N_{1}}i. Also consider the indices

α = (M_{N}, M′_{N}; f_{N}), β = (M_{N_{1}}, M′_{N_{1}}; f_{N_{1}}), γ = (M, M′; f).

Assume p_{γ} = i_{1}^{∗−1}(p_{β}). We have

(i_{1}i)^{∗−1}(p_{α}) = i_{1}^{∗−1}(i^{∗−1}(p_{α})) = i_{1}^{∗−1}(p_{β}),

where the last equality evidently follows from the fact that α < β. Therefore p_{γ} does not depend on the choice of the number N_{1}.

Now consider the diagram

[figure: (M_{N}, M′_{N}); (T_{N}, T′_{N}); (M, M′); (T, T′); (R, R′); φ̃_{1}; φ̃; i; i_{1}; f; f_{1}]

where φ : (M, M′) → (T, T′) is a map such that f_{1}φ = f, φ̃ is a cellular approximation of φ and φ̃_{1} = φ̃|(M_{N}, M′_{N}). We have f ∼ f_{1}φ̃ and (f_{1}i_{1})φ̃_{1} = (f_{1}φ̃)i ∼ fi. Also consider the indices α = (M_{N}, M′_{N}; fi), β = (T_{N}, T′_{N}; f_{1}i_{1}). Applying Lemma 1.2, we have φ^{∗}(i_{1}^{∗−1}(p_{β})) = (φ̃^{∗}i_{1}^{∗−1})(p_{β}) = i^{∗−1}(φ̃_{1}^{∗}(p_{β})) = i^{∗−1}(p_{α}).

We have thus extended the thread of the element p onto the category L_{n}^{(2)}.

Let now (M̄, M̄′) be an arbitrary object of L_{n}^{(1)} and let ḡ : (M̄, M̄′) → (R, R′) be an arbitrary map. Using the standard technique of the homotopy theory, we can, under our assumptions, construct a map φ : (M, M′) → (M̄, M̄′) such that (M, M′) ∈ L_{n}^{(2)} and φ is a homotopy equivalence. Let g = ḡφ. Assume p_{α} = φ^{∗−1}(p_{β}), where α = (M̄, M̄′; ḡ), β = (M, M′; g). We will show that p_{α} does not depend on the choice of φ. Consider the diagram

[figure: (M̃, M̃′); (M, M′); (M̄, M̄′); (R, R′); φ; g; ḡ; φ_{1}; φ̃]

where (M̃, M̃′) ∈ L_{n}^{(2)}, φ̃ and φ_{1} are homotopy equivalences, and φ ∼ φ̃φ_{1}. Let β̃ = (M̃, M̃′; ḡφ̃). Then

g = ḡφ ∼ ḡφ̃φ_{1} = (ḡφ̃)φ_{1}.

Applying Lemma 1.2, we have

φ^{∗−1}(p_{β}) = φ^{∗−1}(φ_{1}^{∗−1}(p_{β})) = φ^{∗−1}(p_{β}).

Consider now an arbitrary map

φ_{0} : (M̄, M̄′) → (T̄, T̄′)

from the category L_{n}^{(1)} and the diagram

[figure: (M, M′); (T, T′); (M̄, M̄′); (T̄, T̄′); (R, R′); φ̃_{0}; φ_{0}; φ; ḡ; ḡ_{1}; φ̃_{1}; φ_{1}]

where ḡ = ḡ_{1}φ_{0}, φ and φ_{1} are homotopy equivalences, φ̃_{1} is the homotopy inverse of φ_{1}, φ̃_{0} = φ̃_{1}φ_{0}φ and (M, M′), (T, T′) ∈ L_{n}^{(2)}. We have (ḡ_{1}φ_{1})φ̃_{0} = ḡ_{1}φ_{1}φ̃_{1}φ_{0}φ ∼ ḡ_{1}φ_{0}φ = ḡφ. Also, consider the indices β = (M, M′; ḡφ), β_{1} = (T, T′; ḡ_{1}φ_{1}). Then, applying Lemma 1.2, we have φ_{0}^{∗}(φ_{1}^{∗−1}(p_{β_{1}})) = (φ_{0}^{∗}φ_{1}^{∗−1})(p_{β_{1}}) = (φ^{∗−1}φ̃_{0}^{∗})(p_{β_{1}}) = φ^{∗−1}(p_{β}).

Let, finally, (X, X′) be an arbitrary object of L_{n}, and let

w_{X} : (S(X), S(X′)) → (X, X′)

be the natural projection of the singular complex S(X) onto X. Let f : (X, X′) → (R, R′) be an arbitrary map. Also, consider the indices α = (X, X′; f), β = (S(X), S(X′); fw_{X}). Assume p_{α} = w_{X}^{∗−1}(p_{β}) and consider the commutative diagram

[figure: (S(X), S(X′)); (S(Y), S(Y′)); (X, X′); (Y, Y′); (R, R′); φ̄; φ; ω_{X}; ω_{Y}; f; g]

where gφ = f and φ̄ is the cellular map induced by φ. We have (gw_{Y})\overline{φ} = gφw_{X} = fw_{X}. Also, consider the indices β_{X} = (S(X), S(X′); fω_{X}), β_{Y} = (S(Y), S(Y′); gω_{Y}). Then

φ^{∗}(ω_{Y}^{∗−1}(p_{β_{Y}})) = ω_{X}^{∗−1}(φ̄^{∗}(p_{β_{Y}})) = ω_{X}^{∗−1}(p_{β_{X}}).

This completes the proof of Theorem 2.1. ■

4. Proof of Theorem 2.4

Consider some full subcategories of F_{n}^{a} (see Section 2):

0) K_{n}^{(0)} = F_{n}^{a};

1) K_{n}^{(1)}-objects are CW -complexes with one vertex and without cells of dimensions 1, 2, …, n − 2;

2) K_{n}^{(2)}-objects are CW -complexes with one vertex and cells of dimensions n − 1, n and n + 1 only;

3) K_{n}^{(3)}-objects are CW -complexes with one vertex and cells of dimensions n − 1 and n only;

4) K_{n}^{(4)} = F_{n}^{b}.

Let R = (R, ∗) be an arbitrary space from K. All subcategories K_{n}^{(i)}, 0 ≤ i ≤ 4, will be regarded as auxiliary ones.

Let

λ_{i} : Π^{n}(R; K_{n}^{(i)}) → Π^{n}(R; K_{n}^{(i+1)}), 0 ≤ i ≤ 3,

be the natural restriction homomorphisms.

Let L′_{n} and L′′_{n} be two small full subcategories of the category K consisting of the spaces (X, ∗), and L′_{n} ⊂ L′′_{n}. It is assumed that the following condition is satisfied: for each X ∈ L′′_{n} there is Y ∈ L′_{n} such that Y has the same homotopy type as X. Consider L′_{n} and L′′_{n} as auxiliary subcategories. Let λ̄ : Π^{n}(R; L′′_{n}) → Π^{n}(R; L′_{n}) be the natural restriction homomorphism.

Lemma 4.1. λ̄ is a natural isomorphism.

Proof. We prove first that the homomorphism λ̄ is a monomorphism. Let p ∈ Π^{n}(R; L′′_{n}), λ̄(p) = 0 and α = (X; f) ∈ ′′ω(R; n) be an arbitrary index. Let Y ∈ L′_{n} and the map φ : Y → X be a homotopy equivalence. Consider the index β = (Y; fφ) ∈ ′ω(R; n) ⊂ ′′ω(R; n). We have β < α. Then φ^{∗}(p_{α}) = p_{β} = [λ̄(p)]_{β} = 0. Therefore p_{α} = 0 and p = 0.

Let us now prove that the homomorphism λ̄ is an epimorphism. Let q ∈ Π^{n}(R; L′_{n}) and α = (X; f) ∈ ′′ω(R; n) be an arbitrary index. Let Y ∈ L′_{n} and the map φ : Y → X be a homotopy equivalence. Consider the index β = (Y; fφ) ∈ ′ω(R; n) and assume p_{α} = φ^{∗−1}(q_{β}). We will show that p_{α} does not depend on the choice of φ. Consider the diagram

[figure: Y; Y_{1}; Y_{1}; Y; X; R; φ̃_{1}φ; φ; φ_{1}; φ̃_{1}; id; id; f; fφ_{1}; fφ]

where Y, Y_{1} ∈ L′_{n}, φ and φ_{1} are homotopy equivalences, and φ̃_{1} is the homotopy inverse of φ_{1}. Consider the index γ = (Y_{1}; fφ_{1}) ∈ ′ω(R; n).

ON COHOMOTOPY-TYPE FUNCTORS 161

Then (fφ_{1})(φ̃_{1}φ) ∼ fφ and therefore β < γ. In this case φ_{1}^{∗}(φ^{∗−1}(q_{β})) = φ_{1}^{∗}(φ^{∗−1}((φ̃_{1}φ)^{∗}(q_{γ}))) =

= (φ_{1}^{∗}φ^{∗−1}φ^{∗}φ_{1}^{∗})(q_{γ}) = (φ_{1}^{∗}φ_{1}^{∗})(q_{γ}) = q_{γ}.

Hence φ^{∗−1}(q_{β}) = φ_{1}^{∗−1}(q_{γ}).

We will show that the set {p_{α}} defines an element of the group Π^{n}(R; L′′_{n}). Consider the diagram

[figure: Y_{0}; Y_{1}; X_{1}; X_{0}; R; φ̃_{1}φφ_{0}; φ_{0}; φ_{1}; φ̃_{1}; φ; f_{0}; f_{1}]

where f_{1}φ = f_{0}, φ_{0} and φ_{1} are homotopy equivalences, φ̃_{1} is the homotopy inverse of φ_{1}, X_{0}, X_{1} ∈ L′′_{n}, and Y_{0}, Y_{1} ∈ L′_{n}. Then

(f_{1}φ_{1})(φ̃_{1}φφ_{0}) ∼ f_{1}φφ_{0} = f_{0}φ_{0}.

Also, consider the indices

α = (X_{0}; f_{0}), β = (X_{1}; f_{1}), α, β ∈ ′′ω(R; n), α_{1} = (Y_{0}; f_{0}φ_{0}), β_{1} = (Y_{1}; f_{1}φ_{1}), α_{1}, β_{1} ∈ ′ω(R; n) ⊂ ′′ω(R; n).

Then α < β, α_{1} < β_{1} and we have φ^{∗}(p_{β}) = φ^{∗}(φ_{1}^{∗−1}(q_{β_{1}})) = φ_{0}^{∗−1}((φ̃_{1}φφ_{0})^{∗}(q_{β_{1}})) = φ_{0}^{∗−1}(q_{α_{1}}) = p_{α}. Finally, let us prove that λ̄(p) = q. Assume that X ∈ L′_{n} ⊂ L′′_{n} and

α = (X; f) ∈ ′ω(R; n) ⊂ ′′ω(R; n).

Define p_{α} by taking φ = id : X → X. Then

[λ̄(p)]_{α} = p_{α} = id^{∗−1}(q_{α}) = q_{α}.

This completes the proof of Lemma 4.1.

As a consequence of the foregoing lemma we have

Proposition 4.2. λ_{0} is a natural isomorphism. Proof. Every CW -complex from K_{n}^{(0)} is homotopically equivalent to some CW -complex from K_{n}^{(1)}.

Proposition 4.3. λ_{1} is a natural isomorphism.

Proof. We will prove in the first place that λ_{1} is a monomorphism. Let p∈Π^{n}(R; K_{n}^{(1)}) and λ_{1}(p) = 0. Consider the index α = (X; f) ∈ ^{(1)}ω(R; n), X ∈ K_{n}^{(1)}, and the diagram

[figure: X^{n+1}; X; R; fi_{X}; f; i_{X}]  (3)

where X^{n+1} is the (n + 1)-skeleton of X and i_{X} is the standard embedding. Let

β = (X^{n+1}; f i_{X}) ∈ ^{(2)}ω(R; n) ⊂ ^{(1)}ω(R; n).

Then β < α, and we have i_{X}^{∗}(p_{α}) = p_{β} = [λ_{1}(p)]_{β} = 0. Therefore p = 0. Assume now that p ∈ Π^{n}(R; K_{n}^{(2)}) and α = (X; f) ∈ ^{(1)}ω(R; n). Consider the diagram (3), the index β, and assume that q_{α} = i_{X}^{∗−1}(p_{β}). We will show that the set {q_{α}} defines an element of Π^{n}(R; K_{n}^{(1)}). Con-

sider the diagram

[figure: X^{n+1}; Y^{n+1}; Y; X; R; φ_{1}; φ; φ̃; i_{X}; i_{Y}; g; f]

where gφ ∼ f, φ̃ is a cellular approximation of φ and φ_{1} = φ̃|X^{n+1}. Then (gi_{Y})φ_{1} = gφ̃i_{X} ∼ fi_{X}. Also, consider the indices

α_{1} = (Y; g) ∈ ^{(1)}ω(R; n),
β_{1} = (Y^{n+1}; gi_{Y}) ∈ ^{(2)}ω(R; n).

We have β < β_{1} and

φ^{∗}(q_{α_{1}}) = φ^{∗}(q_{α_{1}}) = φ^{∗}(i_{Y}^{∗−1}(q_{β_{1}})) = i_{X}^{∗−1}(φ_{1}^{∗}(q_{β_{1}})) = i_{X}^{∗−1}(q_{β}) = q_{α}.

Finally, let us prove that λ_{1}(q) = p.

Consider the index α = (X; f) ∈ ^{(2)}ω(R; n), where X = X^{n+1} ∈ K_{n}^{(2)} ⊂ K_{n}^{(1)}, and the diagram

[figure: X^{n+1}; X; R; f; f; i_{X}]

where i_{X} = id. Then [λ_{1}(q)]_{α} = i_{X}^{∗−1}(p_{α}) = p_{α}.

This completes the proof of Proposition 4.3.

Proposition 4.4. λ_{2} is a natural isomorphism.

Proof. Let q ∈ Π^{n}(R; K_{n}^{(2)}) and λ_{2}(q) = 0. Consider the index α = (P_{n+1}; f) ∈ ^{(2)}ω(R; n), where P_{n+1} ∈ K_{n}^{(2)}. Let P_{n+1}^{(n)} be the n-skeleton of P_{n+1} and i : P_{n+1}^{(n)} → P_{n+1} be the standard embedding. We have the commutative diagram

[figure: P_{n+1}^{(n)}; P_{n+1}; R; fi; f; i]  (4)

Also consider the index

β = (P_{n+1}^{(n)}; fi) ∈ ^{(3)}ω(R; n) ⊂ ^{(2)}ω(R; n).

We have β < α. Let

→ H^{n}(P_{n+1}, P_{n+1}^{(n)}) → H^{n}(P_{n+1}) →^{i^{∗}} H^{n}(P_{n+1}^{(n)}) →

be a part of the cohomological exact sequence for the pair (P_{n+1}, P_{n+1}^{(n)}). Since H^{n}(P_{n+1}, P_{n+1}^{(n)}) = 0, i^{∗} is a monomorphism. Then i^{∗}(q_{α}) = q_{β} = [λ_{2}(q)]_{β} = 0. Therefore q_{α} = 0 and q = 0. Hence λ_{2} is a monomorphism.

Let now q ∈ Π^{n}(R; K_{n}^{(3)}). Consider again the commutative diagram (4) and the corresponding indices α and β. We will prove below that q_{β} ∈ Im i^{∗}. Therefore p_{α} = i^{∗−1}(q_{β}) is the correct definition. Let us show that the set {p_{α}} defines an element p ∈ Π^{n}(R; K_{n}^{(2)}). Consider the diagram

[figure: P_{n+1}^{(n)}; P̄_{n+1}^{(n)}; P̄_{n+1}; P_{n+1}; R; φ_{1}; φ̃; φ; i; i_{1}; f_{1}; f]

where f_{1}φ ∼ f, φ̃ is a cellular approximation of φ and φ_{1} = φ̃|p_{n+1}^{(n)}. Consider the indices

α_{1} = (p̄_{n+1}; f_{1}) ∈ ^{(2)}ω(R; n), β_{1} = (p̄_{n+1}^{(n)}; f_{1}i_{1}) ∈ ^{(3)}ω(R; n).

Since (f_{1}i_{1})φ_{1} = f_{1}φ̃i ∼ fi, we obtain β < β_{1} and therefore φ_{1}^{∗}(q_{β_{1}}) = q_{β}. We have i^{∗−1}(φ_{1}^{∗}(q_{β_{1}})) = φ^{∗}(i_{1}^{∗−1}(q_{β_{1}})) and

φ^{∗}(p_{α_{1}}) = φ^{∗}(p_{α_{1}}) = φ^{∗}(i_{1}^{∗−1}(q_{β_{1}})) = i^{∗−1}(φ_{1}^{∗}(q_{β_{1}})) = i^{∗−1}(q_{β}) = p_{α}.

Finally, we will prove that λ_{2}(p) = q. Let

α = (P_{n+1}; f) ∈ ^{(3)}ω(R; n) ⊂ ^{(2)}ω(R; n)

be an arbitrary index, P_{n+1} ∈ K_{n}^{(3)}. Thus P_{n+1}^{(n)} = P_{n+1}. Then the map i : P_{n+1}^{(n)} → P_{n+1} is the identity map: i = id. Therefore for β = (P_{n+1}^{(n)}; fi) we have β = α. In this case

[λ_{2}(p)]_{α} = p_{α} = i^{∗−1}(q_{β}) = id^{∗−1}(q_{α}) = q_{α}.

It remains to prove that q_{β} ∈ Im i^{∗}. Consider the characteristic map of the CW -complex P_{n+1}

Φ : (C(∨S^{n}), ∨S^{n}) → (P_{n+1}, P_{n+1}^{(n)}),

where C(∨S^{n}) denotes the cone over ∨S^{n} and ∨ denotes the finite one-point union of spaces. Let φ = Φ|(∨S^{n}). Consider the commutative diagram

[figure: ∨S^{n}; P_{n+1}^{(n)}; P_{n+1}; R; fiφ; f; i; φ; fi]

Since iφ ∼ 0, we obtain fiφ ∼ 0. Let γ = (∨S^{n}; fiφ) ∈ ^{(3)}ω(R; n). We have γ < β and q_{γ} = 0 (see the proof of Corollary 1.3). In this case φ^{∗}(q_{β}) = q_{γ} = 0.

Now consider the commutative diagram

[figure: H^{n}(∨S^{n}); 0; H^{n+1}(C(∨S^{n}), ∨S^{n}); 0; H^{n}(P_{n+1}^{(n)}); H^{n+1}(P_{n+1}, P_{n+1}^{(n)}); H^{n}(P_{n+1}); 0; δ; δ; φ^{∗}; Φ^{∗}; i^{∗}]

where Φ^{∗} is an isomorphism. Then, since φ^{∗}(q_{β}) = 0, we have δ(q_{β}) = 0. Therefore q_{β} ∈ Im i^{∗}. This completes the proof of Proposition 4.4.

Lemma 4.5. For each CW -complex P from the category K_{n}^{(3)} there is a CW -complex P̄ from the category K_{n}^{(4)} such that P̄ has the homotopy type of P.

Proof. By the condition H_{i}(P) = 0, i ≠ 0, n−1, n; H_{n−1}(P) ≈ π_{n−1}(P) are finite abelian groups and H_{n}(P) is a finitely generated free abelian group. Thus the group π_{n−1}(P) can be represented in the form

π_{n−1}(P) ≈ Z_{r_{1}} ⊕ Z_{r_{2}} ⊕ ⋯ ⊕ Z_{r_{t}},

where Z_{r_{i}}, i = 1, 2, …, t, are cyclic groups of order r_{i}. Consider the corresponding system ξ_{i} : S^{n−1} → P, 1 ≤ i ≤ t, of generators in the group π_{n−1}(P) and define, by means of ξ_{i}, the map f : ∨_{i=1}^{t}P^{n}(r_{i}) → P. Then f induces isomorphisms in homotopy and homology in dimensions ≤ n − 1. Now consider a system h_{1}, h_{2}, …, h_{s} of generators in the group H_{n}(P). The Hurewicz homomorphism π_{n}(P) → H_{n}(P) for the space P is an epimorphism. In this case we can consider maps φ_{k} : S^{n} → P, k = 1, 2, …, s, such that φ_{k∗}(1) = h_{k}, where 1 ∈ H_{n}(S^{n}). Assume

P̄ = ( ∨_{i=1}^{t} P^{n}(r_{i})) ⋁ ( ∨_{k=1}^{s} P_{k}^{n}(1)),

where P_{k}^{n}(1) = P^{n}(1) = S^{n}, and define by means of the maps f and φ_{k} the map φ = f ∨ (∨φ_{k}) : P̄ → P. Then φ induces isomorphisms of all homology groups. Therefore, under our assumptions, the map φ is a homotopy equivalence. This proves Lemma 4.5.

Lemmas 4.1 and 4.5 imply

Proposition 4.6. λ_{3} is a natural isomorphism.

Propositions 4.2 - 4.4 and 4.6 imply Theorem 2.4.

5. Proof of Theorem 2.5

Let h ∈ H^{n}(Q^{n}) and i_{j,t} : P^{n}(t) → Q^{n} be standard embeddings. Assume

ε(h) = {i_{j,t}^{∗}(h)} ∈ ∏_{j,t} H^{n}(P_{j}^{n}(t)).

Obviously, we have

Lemma 5.1. The correspondence

ε : H^{n}(Q^{n}) → ∏_{j,t} H^{n}(P_{j}^{n}(t))

is an isomorphism.

In the sequel, for convenience, the subcategories F_{n}^{b} and Q_{n} will be denoted by K_{n}^{(4)} and K_{n}^{(5)}. Let R = (R, ∗) be an arbitrary space from K. Let q ∈ Π^{n}(R; K_{n}^{(4)}) and

α = (Q^{n}; f) ∈ ^{(5)}ω(R; n)

be an arbitrary index. Assume f_{j,t} = fi_{j,t} and consider the indices

α_{j,t} = (P^{n}(t); f_{j,t}) ∈ ^{(4)}ω(R; n).

Let

p_{α} = [λ_{4}(q)]_{α} = ε^{−1}({q_{α_{j,t}}}).

We will show that the set {p_{α}} defines an element of the group Π^{n}(R; K_{n}^{(5)}) and the natural isomorphism

λ_{4} : Π^{n}(R; K_{n}^{(4)}) → Π^{n}(R; K_{n}^{(5)}).

By ∨ we will denote the symbol of finite one-point union of spaces. Consider the commutative diagram

[figure: P^{n}(t); ∨_{j,t}P_{j}^{n}(t); Q^{n}; R; f_{k,l}; f; ∨i_{j,t}; ĩ_{k,l}; f̃]

where ĩ_{k,l} and ∨i_{j,t} are standard embeddings, and the indices

α = (Q^{n}; f) ∈ ^{(5)}ω(R; n),
β = (∨_{j,t}P_{j}^{n}(t); f̃) ∈ ^{(4)}ω(R; n),
α_{k,l} = (P^{n}(t); f_{k,l}) ∈ ^{(4)}ω(R; n).

We have (∨i_{j,t})ĩ_{k,l} = i_{k,l}. Therefore α_{k,l} < β. Then

ĩ_{k,l}^{∗}((∨i_{j,t})^{∗}(p_{α})) = i_{k,l}^{∗}(p_{α}) = q_{α_{k,l}}.

Since this is true for arbitrary k and l, we have

(∨i_{j,t})^{∗}(p_{α}) = q_{β}.  (5)

Now consider the diagram

[figure: ∨_{j,t}P_{j}^{n}(t); P^{n}(t); Q^{n}; R; g; f; φ_{1}; ∨i_{j,t}; F; φ]

where fφ ∼ g, F = f(∨i_{j,t}), and (∨i_{j,t})φ_{1} = φ (since P^{n}(t) is a compact space and φ is a continuous map, it follows that there exists a map φ_{1}). Then

F φ_{1} = f( ∨_{j,t} i_{j,t})φ_{1} = fφ ∼ g.

Consider the indices

β = (P^{n}(t), g) ∈ ^{(4)}ω(R; n),
β_{1} = ( ∨_{j,t} P_{j}^{n}(t); F ) ∈ ^{(4)}ω(R; n).

Then β < β_{1}, and by (5) we have

φ^{∗}(p_{α}) = φ_{1}^{∗}((∨_{j,t})^{∗}(p_{α})) = φ_{1}^{∗}(q_{β_{1}}) = q_{β}.

Thus

φ^{∗}(p_{α}) = q_{β}.  (6)

Finally, consider the diagram

[figure: P^{n}(t); Q^{n}; Q^{n}; R; f; f_{1}; φ_{1}; i_{j,t}; φ]

where f_{1}φ ∼ f and fi_{j,t} = φ_{1}. Consider the indices

α_{1} = (Q^{n}; f_{1}) ∈ ^{(5)}ω(R; n),
α_{j,t} = (P^{n}(t); φ_{1}) ∈ ^{(4)}ω(R; n).

Thus α < α_{1}. Therefore by (6) we have

i_{j,t}^{∗}(φ^{∗}(p_{α_{1}})) = (φi_{j,t})^{∗}(p_{α_{1}}) = q_{α_{j,t}}.

Since this equality is true for arbitrary j and t, we obtain φ^{∗}(p_{α_{1}}) = p_{α}. Therefore the set {p_{α}} defines an element p ∈ Π^{n}(R; K_{n}^{(5)}). The map λ_{4} is now defined by setting λ_{4}(q) = p, where q ∈ Π^{n}(R; K_{n}^{(4)}).

168 S. KHAZHOMIA

Let q_{1}, q_{2} ∈ Π^{n}(R; K_{n}^{(4)}). Then we have i_{j,t}^{∗}([λ_{4}(q_{1} + q_{2})]_{α}) = (q_{1} + q_{2})_{α_{j,t}} = (q_{1})_{α_{j,t}} + (q_{2})_{α_{j,t}} = i_{j,t}^{∗}([λ_{4}(q_{1})]_{α})+ +i_{j,t}^{∗}([λ_{4}(q_{2})]_{α}) = i_{j,t}^{∗}([λ_{4}(q_{1})]_{α} + [λ_{4}(q_{2})]_{α}) = i_{j,t}^{∗}([λ_{4}(q_{1}) + λ_{4}(q_{2})]_{α}).

Since this equality holds for arbitrary j and t, we have

λ_{4}(q_{1} + q_{2}) = λ_{4}(q_{1}) + λ_{4}(q_{2}).

Let now φ : S → R be an arbitrary map. Consider the diagram

[figure: Π^{n}(S; K_{n}^{(4)}); Π^{n}(S; K_{n}^{(5)}) ,; Π^{n}(R; K_{n}^{(5)}); Π^{n}(R; K_{n}^{(4)}); λ_{4}; λ_{4}; φ^{#}; φ^{#}]

the element q ∈ Π^{n}(R; K_{n}^{(4)}), indices α, α_{j,t}, and β = φ(α) = (Q^{n}; g) ∈ ^{(5)}ω(R; n),

β_{j,t} = (P^{n}(t); g_{j,t}) ∈ ^{(4)}ω(R; n),

where g = φf, g_{j,t} = gi_{j,t}. Then β_{j,t} = φ(α_{j,t}), and we have

[φ^{#}(λ_{4}(q))]_{α} = [λ_{4}(q)]_{β} = ε^{−1}({q_{β_{j,t}}}).

On the other hand, we have

[λ_{4}(φ^{#}(q))]_{α} = ε^{−1}({[φ^{#}(q)]_{α_{j,t}}}) = ε^{−1}({q_{φ(α_{j,t})}}) = ε^{−1}({q_{β_{j,t}}}). Thus λ_{4} is a natural homomorphism.

We will prove that λ_{4} is a monomorphism. Let q ∈ Π^{n}(R; K_{n}^{(4)}) and λ_{4}(q) = 0. Consider an arbitrary index

β = ( ∨_{j,t} P_{j}^{n}(t); g) ∈ ^{(4)}ω(R; n)

and define the map f : Q^{n} → R by taking

f((∨i_{j,t})(x)) = g(x), x ∈ ∨_{j,t}P_{j}^{n}(t),

f(Q^{n} − (∨i_{j,t})(∨_{j,t}P_{j}^{n}(t))) = ∗.

Consider the index

α = (Q^{n}; f) ∈ ^{(5)}ω(R; n)

and the commutative diagram

[figure: ∨_{j,t}P_{j}^{n}(t); Q^{n}; R; ∨i_{j,t}; g; f]

then by (5) we have

q_{β} = ( ∨_{j,t} i_{j,t})^{∗}([λ_{4}(q)]_{α}) = ( ∨_{j,t} i_{j,t})^{∗}(0) = 0.

Therefore q = 0 and λ_{4} is a monomorphism.

Further, we will prove that λ_{4} is an epimorphism. Let p ∈ Π^{n}(R; K_{n}^{(5)}) and

α = (X; f) ∈ ^{(4)}ω(R; n)

be an arbitrary index, X ∈ K_{n}^{(4)}. Therefore the space X can be represented in the form X = ∨_{j,t}P_{j}^{n}(t), where j is an index indicating a certain arrangement of the identical subspaces of X. Then we have the natural embedding i : X → Q^{n}. Define the map l : Q^{n} → X by taking

l(i(x)) = x, x ∈ X;
l(Q^{n} − i(X)) = ∗.

Let g = fl. We have gi = f. Let

β = (Q^{n}; g) ∈ ^{(5)}ω(R; n).

Assume that

q_{α} = i^{∗}(p_{β}).  (7)

Consider a different arrangement of subspaces of X. Let the map ĩ and the index β̃ = (Q^{n}; g̃) be defined in the same way as i and β, respectively. Consider the commutative diagram

[figure: Q^{n}; Q^{n}; X; R; k; g; g̃; ĩ; i]

where the map k can be defined by a certain permutation of subspaces of Q^{n}. Then β < β̃ and we have

ĩ^{∗}(p_{β}) = i^{∗}(k^{∗}(p_{β})) = i^{∗}(p_{β}).

Thus definition (7) is correct.

Consider the diagram

[figure: X_{1}; X_{2}; R; Q^{n}; Q^{n}; f_{1}; f_{2}; l_{1}; i_{1}; g_{1}; g_{2}; l_{2}; i_{2}; φ; φ̃]

where X_{1}, X_{2} ∈ K_{n}^{(4)}, f_{2}φ ∼ f_{1}, φ̃ = i_{2}φl_{1}, f_{1} = g_{1}i_{1}, f_{2} = g_{2}i_{2}. Then

φ̃i_{1} = i_{2}φl_{1}i_{1} = i_{2}φ,
g_{2}φ̃ = f_{2}l_{2}i_{2}φl_{1} = f_{2}φl_{1} ∼ f_{1}l_{1} = g_{1}.

Consider the indices

α_{t} = (X_{t}; f_{t}) ∈ ^{(4)}ω(R; n), t = 1, 2;
β_{t} = (Q^{n}; g_{t}) ∈ ^{(5)}ω(R; n), t = 1, 2.

Then α_{1} < α_{2}, β_{1} < β_{2} and we have

φ^{∗}(q_{α_{2}}) = φ^{∗}(i_{2}^{∗}(p_{β_{2}})) = i_{1}^{∗}(φ^{∗}(p_{β_{2}})) = i_{1}^{∗}(p_{β_{1}}) = q_{α_{1}}.

Therefore the set {q_{α}} defines an element q ∈ Π^{n}(R; K_{n}^{(4)}).

Finally, let us prove that λ_{4}(q) = p. Consider an arbitrary index

α = (Q^{n}; f) ∈ ^{(5)}ω(R; n)

and the index

α_{j,t} = (P^{n}(t); fi_{j,t}) ∈ ^{(4)}ω(R; n).

Then i_{j,t} = i and we have

i_{j,t}^{∗}([λ_{4}(q)]_{α}) = q_{α_{j,t}} = i_{j,t}^{∗}(p_{α}).

Since this equality is true for arbitrary j and t, we have [λ_{4}(q)]_{α} = p_{α}. Therefore λ_{4}(q) = p. This completes the proof of Theorem 2.5. ■

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(Received 11.09.1992)

Author’s address: A.Razmadze Mathematical Institute Georgian Academy of Sciences 1, Z. Rukhadze St., 380093 Tbilisi Republic of Georgia